I'm not going to spend a whole lot of time describing how they work in general, I'll just differentiate between diesel and gas and point you to some helpful links. 

Gas engine insert air and gas together into the combustion chamber, compress it, then ignite it with a spark from a spark plug.

Diesel engines take in only air, compress it, then at the top of the compression stroke where the air is highly pressurized (about 350-400 psi on average), a diesel injector will squirt a tiny little bit of fuel into the chamber.  Because pressure and heat are so high is creates instant combustion therefore eliminating the need for an auxillary device like a spark plug to ignite the fuel & air mixure.  Diesel is injected either directly into the cylinder or indirectly via a prechamber which first "swirls" the fuel to allow it to adequately combust.

Therefore, the amount of fuel used is greatly less in comparison to a gas engine.  Compression ratios for gas engines are typically between 7:1 to 13:1 air to fuel.  Diesel engines typically operate in the area of 19:1 to 23:1.  With a turbo charger the actual ratio of fuel to air can be increased  becaus the more air you have the more fuel you can burn.

**** One thing I fugured out (as this page brought in the most comments) is that the numbers used above are *compression ratio** statistics, not air to fuel as I had originally indicated.  

In response to this page, Ian Bragg, administrator of the DTLC and 80's Cool email lists had written:

Diesel engines typically operate with air fuel ratios from 100:1 (at idle)
to 22:1 for a NA diesel motor or 18:1for a Turbo diesel motor. Since few
motors operate at full throttle most of their life, diesels are typically
operating at ratios of 25:1 - 50:1, whereas petrol based motors must
maintain a stoichiometric ratio of around 14.7:1 all the time.

A diesel motor being worked hard in a highway situation can return fuel
consumption similar to an equivalent sized EFI petrol engine.

Hope this helps.

Ian B

The numbers I have written above are not representative of idling or variations in the engine's fuel mixture, but are moreso of the number designated as "fuel:air ratio."  For example a 81-85 Volvo 240 has a 10.3:1 ratio and a 2H diesel engine at about 22:1.  In fact, I am wrong, I am quoting compression ratios.
 
 

Here are a few good links:
 How Stuff Work's Diesel Engines
 How Stuff Work's Gas Engines

Other stuff you might want to also check out: 
How Stuff Work's Automotive Section  Very cool!